A quadrature method for a class of strongly oscillatory infinite integrals
Let J(F;\(\lambda\)) be an integral of the form \(J(F;\lambda)=\int^{\infty}_{0}e^{-x}F(x,\lambda x)dx,\) where \(\lambda\) is a large parameter, and F(x,y) is a continuous periodic function of period unity in y, so that F(x,\(\lambda\) x) is rapidly oscillating for large \(\lambda\). A finite sum S(F;\(\lambda\)) is said a quadrature formula for J(F;\(\lambda\)) with compound precision \([p,q,\lambda^{-m}]\), where p, q and m are given positive integers, if the following estimate \(| J(P;\lambda)-S(P;\lambda)| <K\lambda^{-m}\) holds for all compound polynomials P(x,y) of degrees not exceeding p and q in x and y, respectively, of the form \[ P(x,y)=\sum_{0\leq \nu \leq p}\sum_{0\leq \mu \leq q}\{c_{\nu \mu}x^{\nu} \cos (2\pi \mu y)+d_{\nu \mu}x^{\nu} \sin (2\pi \mu y)\}, \] where K is a constant independent of \(\lambda\), and \(c_{\nu \mu}\) and \(d_{\nu \mu}\) are arbitrary coefficients. The authors give a constructive proof that there exists a class of quadrature formulas J(F;\(\lambda\))\(\cong S(F;\lambda)\) with any preassigned compound precision \([p,q,\lambda^{-m}]\) (m\(\geq 3)\).
scientific article; zbMATH DE number 4144658